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| The Cahn-Hilliard equation is discussed with subcritically or some critically growing nonlinearities in $H^1({\mathbb R}^N)$ and a dissipative mechanism is described. This involves a weak form of dissipativeness, in which case each individual solution is suitably attracted by the set of equilibria, and strong dissipativeness, for which we indicate that it cannot be in general expected in $H^1({\mathbb R}^N)$. Two types of perturbations of the Cahn-Hilliard equation are also considered where the dissipative mechanism becomes strong enough to ensure the existence of a compact global attractor.
[1] J.W. Cholewa, A. Rodriguez-Bernal, On the Cahn-Hilliard equation in $H^1({\mathbb R}^N)$, J. Differential Equations 253, 2012, 3678-3726.
[2] J.W. Cholewa, A. Rodriguez-Bernal, A note on the Cahn-Hilliard equation in $H^1({\mathbb R}^N)$ involving critical exponent, preprint. |
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